The most commonly used spline function is the one with $k=3$, the so-called cubic spline function. Assume that we have in adddition to the $n+1$ knots a series of functions values $y_0=f(x_0), y_1=f(x_1), \dots y_n=f(x_n)$. By definition, the polynomials $s_{i-1}$ and $s_i$ are thence supposed to interpolate the same point $i$, that is $$s_{i-1}(x_i)= y_i = s_i(x_i),$$ with $1 \le i \le n-1$. In total we have $n$ polynomials of the type $$s_i(x)=a_{i0}+a_{i1}x+a_{i2}x^2+a_{i2}x^3,$$ yielding $4n$ coefficients to determine.